Problem 20 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
caseworksymmetry
In the triangle WXY, point Z lies on XY and point T lies on WZ, as shown. Connecting T with X creates a figure with nine interior angles. Of those 9 angles, what is the smallest possible number that could all be different sizes from one another?
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Answer: B — 3
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Hint 1 of 3
The drawing makes three small triangles, and each one's angles must add to 180°.
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Hint 2 of 3
Ask how FEW distinct sizes the nine angles could take while still obeying every triangle's angle sum and the straight-line angles at T and Z.
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Hint 3 of 3
Try to force as many angles equal as possible and see what minimum number of different sizes survives.
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Approach: minimise the number of distinct angle sizes under the triangle-sum constraints
- Segment TX cuts the figure into three triangles, and the marked angles also satisfy straight-angle relations along line WZ at T and along XY at Z.
- If we try to make all nine angles equal, the 180° sums clash, so they cannot all match; with care we can still force them down to just a few values.
- Choosing the shape cleverly collapses the nine angles into exactly three distinct sizes, and no arrangement does better.
- So the smallest possible number of different sizes is 3, choice (B).
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